Probing Majorana edge states with a flux qubit
Hou, C-Y; Hassler, F.; Akhmerov, A.R.; Nilsson, J.
Citation
Hou, C. -Y., Hassler, F., Akhmerov, A. R., & Nilsson, J. (2011). Probing Majorana edge states with a flux qubit. Physical Review B, 84(5), 054538. doi:10.1103/PhysRevB.84.054538
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Probing Majorana edge states with a flux qubit
Chang-Yu Hou,1Fabian Hassler,1Anton R. Akhmerov,1and Johan Nilsson2
1Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
2Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden
(Received 13 January 2011; revised manuscript received 24 May 2011; published 15 August 2011) A pair of counterpropagating Majorana edge modes appears in chiral p-wave superconductors and in other superconducting systems belonging to the same universality class. These modes can be described by an Ising conformal field theory. We show how a superconducting flux qubit attached to such a system couples to the two chiral edge modes via the disorder field of the Ising model. Due to this coupling, measuring the backaction of the edge states on the qubit allows us to probe the properties of Majorana edge modes.
DOI:10.1103/PhysRevB.84.054538 PACS number(s): 74.20.Mn, 73.23.−b, 74.50.+r
I. INTRODUCTION
Chiral Majorana fermion edge states were originally pre- dicted to exist in the 5/2 fractional quantum Hall plateau.1 These edge states support not only neutral fermionic excita- tions but also more exotic edge vortices. A single edge vortex corresponds to a π phase shift to all fermions situated to one side of it.2–4 Two edge vortices may either fuse into an edge fermion or annihilate each other, with the outcome depending on the preceding evolution of the system. In other words, the edge theory (together with the corresponding bulk theory) possesses non-Abelian statistics.5–8This unusual physics and its potential applications to topological quantum computation are the reasons why the Majorana edge states have attracted much attention recently.9–14
Similar non-Abelian anyons and their corresponding edge states appear in superconducting systems as well. Initially it was discovered that p-wave superconductors support non- Abelian anyons in the bulk and chiral Majorana edge states.5,15,16Later it was shown that depositing a conventional s-wave superconductor on the surface of a topological insulator while breaking time-reversal symmetry provides an alternative route to realize these non-Abelian states.17–19 Alternative proposals include substituting the topological insulator by a two-dimensional electron gas with spin-orbit coupling20–22or by a half-metal.23,24 The realizations of Majorana edge states using s-wave superconductors have the following advantages:
First, they rely on combining simple, well-studied ingredients.
Second, the materials do not have to be extremely pure, unlike samples needed to support the fractional quantum Hall edge states. Finally, the superconducting implementations of Majorana fermions may feature a larger bulk excitation gap and may therefore be operated at higher temperatures.
The downside of the superconducting implementations of Majorana edge states is the lack of means to manipulate edge vortices.18,19 Different from the 5/2 fractional quantum Hall state, the edge vortices are not coupled to charge and thus cannot be controlled by applying voltages.25 Therefore, the standard proposal to probe the edge vortices in superconduct- ing systems is to inject fermion excitations into the edge, to let them split into edge vortices, and finally to conclude about the behavior of the edge vortices from the detection of the fermion excitations after the subsequent fusion of edge vortices.18,19,25,26
In this paper, we propose a more direct way to manipulate and measure edge vortices using a flux qubit consisting of a superconducting ring interrupted by a Josephson junction.27,28 Our main idea is based on the following observations: First, an edge vortex is created when a superconducting vortex crosses the edge. Second, the motion of the superconducting vortices can be fully controlled by a flux qubit, since by applying a flux bias to the qubit one can tune the energy cost for a vortex being present in the superconducting ring.27 In this way, attaching a flux qubit to a system supporting Majorana edge states allows one to directly create, control, and measure edge vortices without relying on splitting and fusing fermionic excitations.
We note that our proposal is not necessarily advantageous for the purposes of topological quantum computing since quantum computing with Majorana fermions may even be realized without ever using edge states.28–30 Instead the aim of our investigation is to develop a better tool for probing the fractional excitations of the edge theory.
The paper is organized as follows. In Sec.II, we discuss a schematic setup of a system where a pair of chiral Majorana fermion edge modes couples to a flux qubit as a probe of the edge states and briefly list our main findings.
In Sec. III, we review the connection between the one- dimensional critical transverse-field Ising model and Majorana fermion modes. We identify the vortex-tunneling operators between two edge states as the disorder fields of the Ising model, and subsequently derive an effective Hamiltonian for the flux-qubit coupled to Majorana modes. In Sec. IV, we provide the necessary formalism for evaluating the expectation values for the flux-qubit state and qubit susceptibilities. In Sec. V and Sec. VI, we compute the qubit expectation values and the two-point qubit correlation functions in the presence of the edge state coupling, and we use these results to derive the qubit susceptibility. In Sec. VII, we analyze higher-order corrections to correlation functions of the qubit state. We summarize our results in Sec.VIII. Additionally, we provide a brief overview of the flux-qubit Hamiltonian in Appendix A. In Appendix B we reduce the flux-qubit Hamiltonian to that of a two-level system and derive the coupling between the flux-qubit and the Majorana modes. In AppendixC, we give the form of the four point correlation function for the disorder field of the Ising model. Finally, in Appendix D, we provide the detailed derivation of the
FIG. 1. (Color online) Schematic setup of the Majorana fermion edge modes coupled to a flux qubit. A pair of counterpropagating edge modes appears at two opposite edges of a topological superconductor.
A flux qubit, that consists of a superconducting ring and a Josephson junction, shown as a gray rectangle, is attached to the the supercon- ductor in such a way that it does not interrupt the edge states flow.
As indicated by the arrow across the weak link, vortices can tunnel in and out of the superconducting ring through the Josephson junction.
higher order corrections to the correlation functions of the qubit state.
II. SETUP OF THE SYSTEM
In this paper, we consider the following setup: A strip of s-wave superconductor is deposited on the surface of either a three-dimensional topological insulator or a semiconductor with strong spin-orbit coupling and broken time-reversal symmetry (or any other superconducting setup supporting Majorana edge states). As depicted in Fig. 1, a pair of counterpropagating Majorana fermion edge modes appears at the two opposite edges of the superconductor.18,19 To avoid mixing between counterpropagating edge states, the width of the superconductor should be much larger than the superconducting coherence length ¯hvF/. Here and in the following, vF denotes the Fermi velocity of the topological insulator (semiconductor) and is the proximity-induced superconducting pairpotential. To avoid mixing of the two counterpropagating edge modes at the ends of the sample, we require the length of the superconducting strip to be longer than the dephasing length.
A flux qubit, consisting of a superconducting ring with a small inductance interrupted by a Josephson junction, is attached to the heterostructure supporting the Majorana edge modes, as shown in Fig.1. By applying an external flux , the two classical states of the superconducting ring corresponding to the phase difference of 0 and 2π across the junction can be tuned to be almost degenerate.27 In this regime, the flux qubit can be viewed as a quantum two-level system with an energy difference ε (which we choose to be positive) between the states|0 and |2π and a tunneling amplitude δ between them. As described in AppendixA, the energy difference ε can be easily tuned by the external flux threaded through the ring.
The transition between the two qubit states is equivalent to the process of a vortex tunneling through the Josephson junc- tion in or out of the superconducting ring. For convenience, we will refer to the Hilbert space spanned by the qubit states
|0 and |2π as a spin-1/2 system. For example, we are going to call the Pauli matrices σx,y,z acting on the qubit states the qubit spin.
A vortex tunneling through the weak link in the supercon- ductor from one edge to the other is a phase slip of 2π of the superconducting phase difference at the tunneling point. Due to this event, all fermions to one side of the weak link gain a phase of π . As will be shown below, the vortex-tunneling operator can be identified with the operator of the disorder field of a one-dimensional critical Ising model onto which the Majorana edge modes can be mapped.
Since vortex-tunneling events couple the qubit spin to the Majorana edge modes, we expect various observables of the qubit to carry signatures of this coupling. The main theory parameter that we are after is the scaling dimension μ= 1/8 of the edge vortex operator (disorder field). Our main results apply to the regime when vortex tunneling is weak, ε δ.
We find that the reduction of the spin expectation value in the z direction due to the vortex tunneling acquires a nontrivial scaling exponent:
1− σz ∝ δ2
ε2−2μ = δ2
ε7/4. (1)
Similarly, the spin expectation value along the x direction is proportional to ε2μ−1= ε−3/4, thereby probing the scaling dimension of the disorder field.
The finite-frequency susceptibilities that characterize the response of the polarization of the qubit spin to a perturba- tion with frequency ω provide additional information about the Majorana edge states. The susceptibility χzz(ω), which characterizes the change of σz due to a modulation of σz
with frequency ω, is measurable with current experimental techniques. It can be measured by modulating the external flux and reading out the current from a dc-SQUID (SQUID is a superconducting quantum interference device) coupled to the qubit.31,32
The frequency dependence of the susceptibilities exhibits a non-Lorentzian resonant response around the frequency ω≈ ε (here and in the following, we set ¯h= 1). It is modified by the coupling to the Majorana edge states and shows the scaling behavior
|χ(ω)| ∝ 1
|ω − ε|1−2μ = 1
|ω − ε|3/4, (2) as long as ε |ω − ε| and the distance |ω − ε| from the resonance is larger than the width of the resonance. The phase change of susceptibility at the resonance δφ= 3π/4 is different from the π phase change for a usual oscillator. The origin of the extra π/4 phase shift is the Abelian part of the statistical angle of the vortex excitations.9
III. EDGE STATES AND COUPLING TO THE QUBIT A. Coupling of the flux qubit to the edge states The flux qubit has two low-energy states, corresponding to a phase difference φ= 0 or φ = 2π across the Josephson junction at x= x0. The Hamiltonian of the qubit is given by
HQ= −ε 2σz−δ
2eiασ+−δ
2e−iασ−. (3)
The energy difference ε can be tuned by applying an external flux to the qubit while the tunneling amplitude δ > 0 can be manipulated by changing the Josephson coupling of the junction.27 As discussed in Appendix A, the two levels described in Eq. (3) represent the two lowest-energy states localized at the two energy minima of a double-well potential.
For the two-level approximation to be accurate, the energies δ, as well as the driving frequency ω have to be much smaller than the level spacing at each well. The tunneling phase α is proportional to the charge induced on the sides of the junction and its fluctuations are the main source of qubit decoherence.
For simplicity we neglect the charge noise so that we can assume that α is static and set it to zero without loss of generality. The qubit Hamiltonian now reads
HQ= −ε 2σz−δ
2σx. (4)
When there is no phase difference across the Josephson junction (φ= 0), the Hamiltonian of the chiral Majorana modes appearing at the edges of the superconductor, as shown in Fig.1, reads
HMF= ivM 2
dx
2π[ψd(x)∂xψd(x)− ψu(x)∂xψu(x)], (5) where vMis the velocity of the Majorana modes and ψu(x) and ψd(x) are the Majorana fermion fields at the upper and lower edges of the superconductor in Fig. 1. The sign difference between the terms containing ψu and ψd is due to the fact that the modes are counterpropagating. The Majorana fermion fields obey the anticommutation relations
{ψu(x),ψu(x)} = {ψd(x),ψd(x)} = 2πδ(x − x), {ψu(x),ψd(x)} = 0. (6) A vortex tunneling through the weak link at x= x0
advances the phase of each Cooper pair in the region x x0
by 2π . For Majorana fermions, just like any other fermions, this results in a phase shift of π . The effect of this phase shift is a gauge transformation
HMF→ P HMFP , (7)
where the parity operator P is given by P = exp
iπ
x0
−∞dx ρe(x)
, (8)
with the fermion density ρe(x)= ψ†(x)ψ(x) and ψ= (ψu+ iψd)/2√
π. We refer to Appendix B for a derivation of the qubit Hamiltonian and the gauge transformation in Eq. (7).
When the phase difference between two sides of the Josephson junction is exactly π , the Majorana modes approaching the junction are fully reflected.17 Since this phenomenon occurs only very close to the phase difference of π , where the system spends only a short amount of time during the process of a phase slip, we will neglect the effect of this backscattering.
The relation between the phase slip and the parity operator was discussed and used in previous work focusing on the 5/2 fractional quantum Hall state.3,4,14
Combining the Hamiltonian of the Majorana edge states in Eqs. (5) and (7) with the qubit Hamiltonian in Eq.(4), we get
the full Hamiltonian of the coupled system in the basis of|0
and|2π:
H =
HMF 0
0 P HMFP
+ HQ. (9)
The first part of the Hamiltonian represents the chiral Majorana edge states coupled to the phase slip of the superconductor while the second part is the bare flux-qubit Hamiltonian.
Because the parity operator of Eq. (8) is highly nonlocal if expressed in terms of Majorana fermions, it is desirable to map the Majorana modes on a system where the vortex-tunneling event becomes a local operator. To this end, we establish the equivalence of the chiral Majorana edge modes with the long- wavelength limit of the one-dimensional transverse-field Ising model at its critical point.33,34
B. Mapping on the critical Ising model
The lattice Hamiltonian of the Ising model at the critical point is given by33,34
HI = −J
n
snxsxn+1+ szn
, (10)
where snαare the spin-1/2 operators at site n. With the Jordan- Wigner transformation,
sn+= cnexp
⎛
⎝iπ
j <n
c†jcj
⎞
⎠ ,
(11) sn−= c†nexp
⎛
⎝iπ
j <n
c†jcj
⎞
⎠ , snz= 1 − 2c†ncn,
the Ising model of Eq. (10) can be cast in terms of fermions as HI = J
n
[(cn− cn†)(cn+1+ c†n+1)+ c†ncn− cncn†]. (12) Here si±≡ (six± isiy)/2 obey the usual on-site spin commu- tation relations while the fermions operators c†i and ci obey canonical anticommutation relations.
For each fermion, we introduce a pair of Majorana operators ψn= ψn†and ¯ψn= ¯ψn†such that
cn=e−iπ/4
2 (ψn+ i ¯ψn). (13) The Majorana fermions satisfy the Clifford algebra:
{ψm,ψn} = { ¯ψm, ¯ψn} = 2δmn, {ψm, ¯ψn} = 0. (14) In terms of the Majorana operators, the Hamiltonian of Eq. (12) assumes the form
HI = −iJ 2
n
(ψnψn+1− ¯ψnψ¯n+1
+ ψnψ¯n+1− ¯ψnψn+1− 2ψnψ¯n). (15) In the long-wavelength limit, the Hamiltonian of Eq. (15) reduces to Eq. (5) with the identification of the continuum Majorana operators,
ψu(x)→
π
aψn, ψd(x)→
π
aψ¯n, x → na, (16)
and the velocity vM → 2J a. To complete the mapping, the bandwidth of the Ising model should be related to the cutoff energy of the linear dispersion of the Majorana edge states,
→ J . Thereby, a pair of counterpropagating Majorana edge states, ψu(x) and ψd(x), can be mapped on the low energy sector of the one-dimensional transverse-field Ising model at its critical point.
For the parity operator in Eq. (8), we obtain a representation in terms of the Ising model with the following procedure:
We first discretizex0
dx ρe(x) using the mapping of Eq. (16) and identify x0≡ n0a as a lattice point on the Ising model.
Thereafter, we obtain an expression for the vortex tunneling operator P in terms of the Ising model,
P → exp
⎛
⎝iπ
jn0
cj†cj
⎞
⎠ =
jn0
sjz≡ μxn0+1/2, (17)
by using Eq. (13) and the Jordan–Wigner transformation of Eq. (11). Here, μx is the disorder field of the Ising model, i.e., the dual field of the spin field.33–36The Ising Hamiltonian has a form identical to Eq. (10) when expressed through μ operators,
HI = −J
n
μxn−1/2μxn+1/2+ μzn+1/2
, (18)
with μzn+1/2= snzsnz+1.37 We see that the parity operator is indeed a local operator in the dual description of the Ising model. After mapping on the Ising model, Eq. (7) becomes (here and in the following, we use the shortcut notation μ= μx)
P HMFP → μn0+1/2HIμn0+1/2, (19) and the full Hamiltonian of Majorana edge states and the flux qubit of Eq. (9) maps onto
H → HI =
HI 0
0 μn0+1/2HIμn0+1/2
+ HQ. (20) Finally, an additional unitary transformation,
HI → V HIV†, (21)
V = V†=
1 0
0 μn0+1/2
, (22)
yields
HI = HI −ε 2τz−δ
2τxμn0+1/2. (23) Here, τi are the Pauli matrices acting in the Hilbert space spanned by |0 and μn0+1/2|2π. The operators of the qubit spin can be expressed through τx,y,zas
σz= τz, σx = τxμn0+1/2, σy= τyμn0+1/2. (24) We use the Hamiltonian in the form of Eq. (23) and the qubit spin operators of Eqs. (24) in the rest of the paper.
The way of identifying two edge Majorana states with the complete transverse-field Ising model presented above is different from the one commonly used in preceding research.
Usually, the chiral part of the Ising model is identified with a single Majorana edge.2,25 The advantages of our method are
the possibility to write a complete Hamiltonian of the problem and simplified bookkeeping, while its drawback is the need for the right-moving edge and the left-moving edge to have the same geometries. Overall the differences are not important, and both methods can be used interchangeably.
IV. FORMALISM
To probe the universal properties of Majorana edge states, the energy scales of the qubit should be much smaller than the cutoff scale of the Ising model, ε, δ . In the weak coupling limit ε δ, we construct a perturbation theory in δ/ε by separating the HamiltonianHI = H0+ V into an unperturbed part and a perturbation
H0= HI −ε
2τz, V = −δ
2τxμ. (25) Without loss of generality we set ε > 0, so that the ground state of the unperturbed qubit is|0. For brevity we omit the spatial coordinate of the μ operator in the following since it is always the same in the setup that we consider.
We use the interaction picture with time-dependent opera- tors
O(t) = eiH0tOe−iH0t. (26) The perturbation V (t) in this picture is given by
V(t)= −δ
2μ(t)[τ+(t)+ τ−(t)], (27) where τ±(t)= e∓iεtτ± are the time-dependent raising and lowering operators. The structure of the raising and lowering operators leads to physics similar to the Kondo and Luttinger liquid resonant tunneling problems.2,38,39
In the calculation we need the real-time two-point and four- point correlation functions of μ in the long-time limit |t − t| 1. The two-point correlation function is
μ(t)μ(t) = e−isgn(t−t)π/8
2μ|t − t|2μ, (28) where sgn(x) denotes the sign of x and μ= 1/8 the scaling dimension of the μ field.40The phase shift π/8 of the two-point correlator is the Abelian part of the statistical angle for the Ising anyons braiding rules.9Correlation functions involving a combination of multiple fields can be obtained via the underlying Ising conformal field theory or via a bosonization scheme.40–42 The expression for the four-point correlation function is given in AppendixCdue to its length. For brevity we will measure energies in units of and times in units of 1/ in the following calculation and restore the dimensionality in the final result.
We are interested in observables of the flux qubit: the spin expectation values and the spin susceptibilities. We use time- dependent perturbation theory to calculate these quantities.43 This method is straightforward because of the simple form of the perturbing Hamiltonian of Eq. (27) in terms of raising and lowering operators.
Assuming that the system is in the unperturbed ground state at time t0→ −∞, the expectation value of a qubit spin operator σα(t) is expressed through the S matrix S(t,t),
σα(t) = S(t,t0)†σα(t)S(t,t0)0, (29) S(t,t)= T exp
−i
t t
V(s)ds
, t > t. (30) Here,T is the time-ordering operator and ·0is the expectation value with respect to the unperturbed ground state. Similarly, the two-point correlation functions of the qubit spin are given by
σα(t)σβ(0) = S†(t,t0)σα(t)S(t,0)σβ(0)S(0,t0)0. (31) The perturbative calculation for both the expectation values and correlation functions is done by expanding the S matrices in V order by order. This procedure is equivalent to the Schwinger–Keldysh formalism, with the expansion of S and S†corresponding to insertions on the forward and backward Keldysh contours.
According to linear response theory, the susceptibility is given by the Fourier transform of the retarded correlation function of the qubit:43
χαβ(ω)= i
∞
0
dt eiωt[σα(t),σβ(0)]c
= −2
∞
0
dt eiωtImσα(t)σβ(0)c, (32) where·cdenotes the cumulant,
σα(t)σβ(0)c= σα(t)σβ(0) − σα(t)σβ(0), (33) and we have usedσβ(0)σα(t)c= σα(t)σβ(0)∗c. We see that to calculate the susceptibilities only the imaginary part of the correlation functions for t > 0 is required.
V. EXPECTATION VALUES OF THE QUBIT SPIN In this section, we calculate the expectation values of the qubit spin due to coupling with the Majorana edge states to the lowest nonvanishing order. Using the identity
σz= 1 − 2σ−σ+, (34)
we obtain
σz − σz(0)= −2σ−σ+ = −2τ−τ+, (35) sinceσz(0)= 1.
The first nonvanishing correction in the perturbative calcu- lation ofσ−σ+ is of second order in V . By expanding S and S†in Eq. (29), we obtain
τ−τ+(2)=
0
−∞
dt1
0
−∞
dt2Iz,
where
Iz= V (t2)τ−τ+V(t1)0. (36) The integrand Izoriginates from the first-order expansion of both S and S†. The second order contributions from the same S or S† matrix vanish due to the structure of V in the qubit spin space.
Substituting Eqs. (27) and (28) into the integrand Izyields Iz=δ2eiε(t1−t2)−isgn(t2−t1)π/8
4|t2− t1|2μ . (37) By evaluating the integral in Eq. (36), we find
σz(2)= −2τ−τ+(2)= −3(34)δ2
8ε2−2μ, (38) where (x) denotes the gamma function.
The expectation value of σxin the unperturbed ground state vanishes. The first nonvanishing contribution toσx arises to first order in δ/ε. Expanding S and S†in Eq. (29) to first order yields
σx(1)=
0
−∞
dt1Ix,
where
Ix = −i[τxμ(0),V (t1)]0 =sin(−εt1+π8)δ
|t1|2μ , (39) after substituting σx from Eqs. (24) and employing the two point correlator, Eq. (28). Evaluating Eq. (39), we find
σx(1)= (34)δ
ε1−2μ. (40)
Finally,σy = 0 to all orders in perturbation theory since the Hamiltonian is invariant under σy → −σy.
VI. CORRELATION FUNCTIONS AND SUSCEPTIBILITIES OF THE FLUX-QUBIT SPIN Since we are interested in the behavior of susceptibilities at frequencies close to the resonance ω≈ ε, we need to obtain only the long-time asymptotic of the correlation functions of the qubit spine. Using Eqs. (24) and (28), we immediately obtain that
σx(t)σx(0)c=e−iεt−iπ/8
t2μ (41)
is nonvanishing to zeroth order. This is due to the fact that flipping the qubit spin automatically involves creation of an edge vortex, and σx is exactly the spin-flip operator. In the same manner, one obtains thatσy(t)σy(0)c= σx(t)σx(0)c
to zeroth order.
Concentrating next on the mixed correlator, relations (24) and (34) yield
σx(t)σz(0)c= −2μ(t)τx(t)τ−(0)τ+(0)0. (42) The leading nonvanishing term in this correlation function is of first order in δ and given by
σx(t)σz(0)(1)c = −δ
εσx(t)σx(0)c (43) in the long-time limit.
The leading-order contribution to σz(t)σz(0)c can be evaluated using Eq. (28) with expansions of S and S†to second order in δ. In the long-time limit, the leading contribution of the correlation function is given by
σz(t)σz(0)(2)c = δ2
ε2σx(t)σx(0)c. (44)
Correlators containing a single σy vanish because of the invariance under σy→ −σy. We see that all the nonvanishing two-point correlation functions are the same up to overall prefactors. Therefore, we will focus onσx(t)σx(0)c in the following.
A. Energy renormalization and damping
The coupling of the flux qubit to the continuum Majorana edge states can be thought of as a two-level system coupled to an environment via interaction (27). This coupling leads to self-energy corrections for the qubit Hamiltonian,
H0→ H0+ , =
↑↑ ↑↓
↓↑ ↓↓
, (45)
that effectively shifts the energy spectrum and can also induce damping.44Since we are interested only in qubit observables, we focus on the structure of for the two-level system and do not discuss the self-energy correction of the Majorana edge states.
To second order, the self-energy correction for two spin states can be written in terms of the perturbed Hamiltonian of Eqs. (25) as44
αβ= α; 0|V + V (Eα+ i0+− H0)−1V|0; β, (46) where Eα is the energy for the spin-α=↑,↓ qubit states and |α; 0 indicates that the Ising model is in its ground state with spin α for the qubit state. Due to the structure of the Hamiltonian of Eqs. (25), the first order correction to the self-energy vanishes. Additionally, the off-diagonal self-energy corrections vanish, also to second order.
By inserting a complete set
EI,β|EI; ββ; EI| = 1 of the Hilbert space ofH0, with EI denoting the complete set of eigenstates with energy EI for the Ising sector, the diagonal elements of the self-energy become
αα =
EI,β
α; 0|V |EI; ββ; EI|V |0; α
Eα+ i0+− (EI + Eβ) . (47) Because V = −(δ/2)τxμ, only terms with α= β give nonva- nishing contributions such that
αα = δ2 4
EI
0|μ|EIEI|μ|0
±ε − EI + i0+ , (48) where+ corresponds to α =↓, and − to α =↑. The diagonal elements of the self-energy in Eq. (48) can be cast in the form
αα= −iδ2 4
∞
0
dt e±iεte−0+tμ(t)μ(0). (49) To see that Eq. (49) is equal to Eq. (48), we first insert a complete set of states of the Ising model, then write the time evolution of μ in the Heisenberg picture, and finally evaluate the integral.
Evaluating Eq. (49) with Eq. (28) yields
↑↑= − δ2(43)
4ε1−2μ, ↓↓= e−iπ/4 δ2(34)
4ε1−2μ, (50) where we have used ε > 0. The absence of the imaginary part for ↑↑ indicates that the spin-up state is stable. The
self-energy thus gives an energy shift to the spin-up state while it gives an energy shift with a damping to the spin-down state,
Eα= ±ε 2 → ±ε
2 + αα. (51)
The energy renormalization and damping in Eq. (51) alter the time evolution of the ground-state correlation function
τ+(t)τ−(0)0= e−iεt→ e−i(ε+ν)t−γ t/2, (52) where the energy renormalization and damping ν− iγ /2 ≡
↓↓− ↑↑are given by ν= cos2(π8)(34)δ2
2ε1−2μ , γ = 3
4
δ2 2√
2ε1−2μ. (53) At zero temperature, this correlator is the only nonvanishing qubit correlator that enters in the perturbative calculation.
Therefore, the effect of the self-energy can be captured by replacing
ε→ ε + ν − i
2γ , (54)
in the qubit correlation functions computed in the long- time limit, excluding the self-energy correction. Using the replacement rule of Eq. (54), one obtains the zero- temperature correlator
σx(t)σx(0)c=e−i(ε+ν)t−γ t/2−iπ/8
t2μ . (55)
The energy renormalization and the induced damping of Eq. (51) do not arise explicitly in the lowest-order perturbation and require the resummation of the most divergent contribu- tions to all orders in perturbation theory. In a system where Wick’s theorem applies, the resummation for the self-energy can be derived explicitly from a diagrammatic perturbation scheme.43Because the correlation functions of multiple μ’s do not obey Wick’s theorem (see AppendixC), the resummation procedure for our system becomes more complicated. In the long-time limit, however, the most divergent contributions in all orders can be collected by using the operator product expansion for two μ fields that resembles the structure of Wick’s theorem.41,42
B. Finite temperature
Besides γ , finite temperature is an alternative source of decoherence. The finite-temperature correlators of disorder fields are readily obtained from the zero-temperature correla- tors using a conformal transformation,45
1
t2μ → (π kBT)2μ
[sinh(π kBT t)]2μ, (56) where T denotes temperature and kB the Boltzmann constant.
The finite-temperature correlator σx(t)σx(0)c in the long- time limit can be obtained by substituting Eq. (56) into Eq.
(55) with the proviso ε kBT such that the temperature has no direct effect on the qubit dynamics.
C. Susceptibility
With the correlation functions derived above, we are now in the position to evaluate susceptibilities of the qubit. We should keep in mind that these correlators are valid only in the
long-time limit and only be used to study the behavior of only the susceptibilities close to the resonant frequency ω≈ ε.
Evaluating Eq. (32) with Eq. (55) yields the susceptibility at zero temperature around the resonance,
χxx(ω)= ei3π/83
4
[i(ε+ ν − ω) + γ /2]1−2μ, (57) where ν and γ are given in Eqs. (53). Here, we note that the susceptibility of Eq. (57) shows non-Lorentzian response.
This is in contrast to the conventional Lorentzian response of a two-level system weakly coupled to the environment.44,46If we neglect ν and γ , which are of higher order in δ/ε, this susceptibility reduces to
χxx(ω)= (34)
|ω − ε|1−2μ
1, for ω < ε
ei3π/4, for ω > ε,, (58) so it diverges and changes the phase by 3π/4 at the resonant frequency. We can attribute this phase change to the phase shift of the correlator of two disorder fields in Eq. (28).
The presence of damping γ in Eq. (57) provides a cutoff for the divergence of the response on resonance. The maximal susceptibility is reached at ω= ε + ν, and its value is given by
|χxx(ε+ ν)| = 21−2μ3
4
γ1−2μ . (59)
Using the proportionality of correlation functions (43) and (44), one gets that χxz= χzx = −(δ/ε)χxx and χzz= (δ/ε)2χxx. It is interesting to note that when δ→ 0 both χxxand χxzare divergent while χzzvanishes at the resonance.
In Fig.2, the absolute value of the susceptibility|χxx(ω)|
close to the resonance is plotted as a function of frequency. The dotted line shows the modulus of Eq. (58) for ν= γ = 0 while the dashed line shows that of Eq. (57). A renormalization of the resonant frequency ν becomes clearly visible when comparing the peak position of the dashed line with that of the dotted line.
0 2
1
ω/ε
T = 0, δ = 0 T = 0, δ = 0 T = 0, δ = 0
|χ(ω)|γ3/4Λ1/4
1.2
0.9 1.1
8 .
0 1.0
FIG. 2. Plot of the magnitude of the susceptibility|χxx(ω)| as a function of frequency ω close to resonance ε. The dotted line shows the zero-temperature susceptibility in the absence of the damping and energy renormalization while the dashed line shows the result in the presence of the energy shift and the damping in Eq. (51).
The parameters used for the plot are ε= 0.1 and δ/ε = 0.2. The solid line shows a plot of the finite-temperature susceptibility with kBT = 0.02ε.
The conformal dimension of the vortex excitation can be measured in the region with ε |ω − ε| γ where
|χxx(ω)| = 3
4
|ω − ε|1−2μ. (60) Moreover, both χxzand χzzexhibit the same scaling behavior.
The finite-temperature susceptibility of χxx(ω,T ) can be evaluated from correlation function (55) subjected to the transformation of Eq. (56). The result is plotted as the solid line in Fig.2. An immediate effect of the temperature is that it also introduces a cutoff for the divergence on resonance.
For instance, the resonance peak of the susceptibility yields a different scaling behavior with respect to the temperature,
|χxx(ε+ ν,T )| ∝ T−(1−2μ), (61) as long as π kBT γ . The zero-temperature scaling behavior of the resonance peak in Eq. (59) will be masked by a finite temperature with a crossover at π kBT ≈ γ . These scaling and crossover behaviors of the resonance strength are features of the coupling of the Majorana edge states and the flux qubit.46
The finite-temperature susceptibility shows a resonance at ε+ ν, as shown in Fig.2. Around the resonance, the frequency dependence at finite temperature will be given by the power law in Eq. (60) but with the region constrained by π kBT instead of γ if π kBT > γ.
VII. HIGHER-ORDER CORRELATOR
So far, we have computed the qubit susceptibilities to their first nonvanishing orders and the lowest-order self-energy correction ε→ ε + ν − iγ /2. As a consequence, we used only the two-point correlation functions μ(t)μ(0) in our evaluations. The next nontrivial corrections to the qubit correlators involve the equal-position four-point correlator of the disorder fieldsμ(t1)μ(t2)μ(t3)μ(t4). As discussed in AppendixC, the four-point correlator, in principle, contains information about the non-Abelian statistics of the particles because changing the order of the fields in the correlation function not only alters the phase but can also change the functional form of the correlator.14 It is thus interesting to go beyond the lowest nonvanishing order. Additionally, doing so allows us to check the consistency of the calculation of the self-energy correction done in Sec.VI A.
As an example we focus on the second-order correction to theσx(t)σx(0)ccorrelator in the long-time limit. The details of the calculation are given in AppendixDand the result in Eq. (D33). The dominant correction is a power-law divergence,
σx(t)σx(0)(2)c ∝ e−iπ/8e−iεt t2μ
1−
iν+γ 2
t
, (62) which is just the second order in the δ expansion of the modified correlation function
σx(t)σx(0)c∝ e−i(ε+ν)te−γ t/2e−iπ/8
t2μ . (63)
Hence, we confirm that the second-order perturbative cor- rection is consistent with the the self-energy correction calculation.
The leading correction to the susceptibility χxx in second order is due to the logarithmic term ∝ t−1/4log t in the correlator in Eq. (D33) and has the form
χxx(2)(ω)= − δ2(2+√ 2)7
4
3
4
ei3π/8 16ε7/4[i(ε+ ν − ω) + γ /2)]1−2μ
× ln
(γ /2)2+ (ω − ε − ν)2 ε2
(64) where we have included the self-energy correction of Eq.
(54), and omitted terms without logarithmic divergence.
Unfortunately the effects of nontrivial exchange statistics of disorder fields are not apparent in this correction.
VIII. CONCLUSION AND DISCUSSION
We have proposed a scheme to probe the edge vortex excitations of chiral Majorana fermion edge states realized in superconducting systems utilizing a flux qubit. To analyze the coupling we mapped the Hamiltonian of the Majorana edge states on the transverse-field Ising model, so that the coupling between the qubit and the Majorana edge modes becomes a local operator. In the weak coupling regime δ ε we have found that the ground-state expectation values of the qubit spin are given by
σx = 3
4
δ
ε1−2μ2μ, σy = 0, σz = 1 −3δ
8εσx. (65) Additionally, the susceptibility tensor of the qubit spin in the basis x,y,z is given by
χ(ω)= χxx(ω)
⎛
⎜⎝
1 0 −δ/ε
0 1 0
−δ/ε 0 (δ/ε)2
⎞
⎟⎠, (66)
χxx(ω)= ei3π/83
4
[i(ε+ ν − ω) + γ /2]1−2μ2μ, (67) with the real part ν and the imaginary part γ /2 of the self- energy given by
ν=cos2π
8
3
4
δ2
2ε1−2μ2μ , γ /2= (√
2− 1)ν. (68) We see that all of these quantities acquire additional anomalous scaling (ε/)2μdue to the fact that each spin flip of the qubit spin couples to a disorder field μ. Similar scaling with temperature appears in interferometric setups,25but using a flux qubit allows us to attribute its origin to the dynamics of vortices much more easily and also gives additional tunability of the strength of the coupling. Another effect of the vortex tunneling being present is the phase change δφ= 3π/4 of the susceptibility around the resonance.46This phase shift occurs due to the anomalous scaling and the presence of the Abelian statistical angle of the disorder field, in view of the fact that χxxis just a correlator of two disorder fields in the frequency domain.
The long-wavelength theory which we used is applicable only when all of the energy scales are much smaller than the cutoff energy of the Majorana modes. This is an important constraint for the flux qubit coupled to the Majorana edge states. In systems where the time-reversal symmetry is broken
in the bulk (unlike for topological insulator-based proposals47), the velocity of the Majorana edge states can be estimated to be vM ∝ vF/EF and the dispersion stays approximately linear all the way up to . The cutoff of the Majorana modes is related to the energy scale of the Ising model = → J . Equating J = and vM = 2J a, we obtain the lattice constant of the Ising model a= vF/EF ≡ λF, with λFthe Fermi wavelength.
The Fermi wavelength is typically smaller than any other length scale, and so the long wavelength approximation we have used is well justified. For a typical flux qubit the tunneling strength δ is indeed much smaller than the superconducting gap; the level splitting ε may vary from zero to quantities much larger than the superconducting gap.
Our proposal provides a way to measure properties of the non-Abelian edge vortex excitations different from the conventional detection scheme that requires fusing vortices into fermion excitations. However, none of our results for the single flux qubit can be directly connected to the non-Abelian statistics of the quasiparticles, even after including higher- order corrections. Thus, it is of interest for future research to investigate a system where the edge vortex excitations are coupled to two qubits such that braiding of vortex excitations can be probed.9Another feature of systems with several qubits worth investigating is the ability of the Majorana edge modes to mediate entanglement between different flux qubits.
ACKNOWLEDGMENTS
We thank C. W. J. Beenakker for useful discussions. This research was supported by the Dutch Science Foundation NWO/FOM (C.-Y. H., A. A., and F. H.) and the Swedish Research Council (Vetenskapsr˚adet) (J. N.).
APPENDIX A: FLUX QUBIT
The flux qubit which we consider consists of a supercon- ducting ring interrupted by a Josephson junction which is parameterized by its critical current Ic, its capacitance C, and the self-inductance L of the ring threaded by a magnetic flux
. The Hamiltonian in the phase basis reads27 H = −4EC
d2
dφ2 + EJ(1− cos φ) +EL
2 (φ− 2π/0)2, (A1) where φ is the phase difference across the Josephson junc- tion and 0= h/2e is the superconducting flux quantum.
We have introduced the charging energy EC = e2/2C, the Josephson energy EJ = 0Ic/2π , and the inductive energy EL= 20/4π2L.
The potential energy is given by the last two terms of the Hamiltonian of Eq. (A1). Neglecting for a moment the inductive energy, the cosine potential favors states with φ= 2π Z. The transition between these state involves a change of the phase difference by 2π which corresponds to driving a vortex in or out of the superconducting loop. The inductive energy breaks the degeneracy of the states with a different number of vortices n in the loop by favoring states with n0≈ . When the flux is tuned close to 0/2, the system becomes frustrated since the states φ= 0 and φ = 2π are then nearly degenerate in energy. When the inductive energy
